Related papers: On ternary Egyptian fractions with prime denominat…

In the present paper we prove that every sufficiently large odd integer $N$ can be represented in the form \begin{equation*} N=p_1+p_2+p_3\,, \end{equation*} where $p_1,p_2,p_3$ are primes, such that $p_1=x^2 + y^2 +1$, $p_2=[n^c]$.

We consider the sum of the reciprocals of the middle prime factor of an integer, defined according to multiplicity or not. We obtain an asymptotic expansion in the first case and an asymptotic formula involving an implicit parameter in the…

A new formula for the partition function $p(n)$ is developed. We show that the number of partitions of $n$ can be expressed as the sum of a simple function of the two largest parts of all partitions. Specifically, if $a_1 + >... + a_k = n$…

Let $a$ and $b$ be positive integers and let $p$ be an odd prime such that $p=ax^2+by^2$ for some integers $x$ and $y$. Let $\lambda(a,b;n)$ be given by $q\prod_{k=1}^\infty (1-q^{ak})^3(1-q^{bk})^3 = \sum_{n=1}^\infty \lambda(a,b;n)q^n$.…

Let $X$ be a sufficiently large positive integer. We prove that one may choose a subset $S$ of primes with cardinality $O(\log X)$, such that a positive proportion of integers less than $X$ can be represented by $x^2 + p y^2$ for at least…

Let $p$ be a prime, $k$ a positive integer and let $\mathbb{F}_q$ be the finite field of $q=p^k$ elements. Let $f(x)$ be a polynomial over $\mathbb F_q$ and $a\in\mathbb F_q$. We denote by $N_{s}(f,a)$ the number of zeros of…

Let $P(m)$ denote the largest prime factor of an integer $m\geq 2$, and put $P(0)=P(1)=1$. For an integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq 2-k}$ be the $k-$generalized Fibonacci sequence which starts with $0,...,0,1$ ($k$ terms) and…

We investigate the Waring-Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers. Define $s_k=2k(k-1)$ when $k\ge 3$, and put $s_2=6$. In addition, put $\theta_2=\frac{19}{24}$,…

Let $n,p,k$ be three positive integers. We prove that the rational fractions of $q$: $${n \brack k}_{q} {}_3\phi_{2} [ . {matrix}q^{1-k},q^{-p},q^{p-n} q,q^{1-n} {matrix}| q;q^{k+1}]\quad\textrm{and}\quad q^{(n-p)p}\qbi{n}{k}{q} {}_3\phi_2[…

We show that for Beurling generalized numbers the prime number theorem in remainder form $$\pi(x) = \operatorname*{Li}(x) + O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all } n\in\mathbb{N}$$ is equivalent to (for some $a>0$) $$N(x) =…

Let $\mathcal{P}$ denote the set of all primes. $P_{1},P_{2},P_{3}$ are three subsets of $\mathcal{P}$. Let $\underline{\delta}(P_{i})$ $(i=1,2,3)$ denote the lower density of $P_{i}$ in $\mathcal{P}$, respectively. It is proved that if…

We present some congruences modulo $p^{6-d}$ for sums of the type $\sum_{k=0}^{(p-3)/2}x^k{2k\choose k}/(2k+1)^d$, for $d=1,2,3$ where $p>5$ is a prime.

The ternary Goldbach conjecture (or three-prime conjecture) states that every odd number greater than 5 can be written as the sum of three primes. The purpose of this book is to give the first proof of the conjecture, in full.

Given a polynomial $f(x_1,x_2,\ldots, x_t)$ in $t$ variables with integer coefficients and a positive integer $n$, let $\alpha(n)$ be the number of integers $0\leq a<n$ such that the polynomial congruence $f(x_1, x_2, \ldots, x_t)\equiv a\…

For integers a and n>0, let a(n) denote the residue class {x\in Z: x=a (mod n)}. Let A be a collection {a_s(n_s)}_{s=1}^k of finitely many residue classes such that A covers all the integers at least m times but {a_s(n_s)}_{s=1}^{k-1} does…

A sequence $A$ of positive integers having the property that no element $a_i \in A$ divides the sum $a_j+a_k$ of two larger elements is said to have `Property P'. We construct an infinite set $S\subset \mathbb{N}$ having Property P with…

Let $n$ be a positive integer. In this paper we estimate the size of the set of linear forms $b_1\log a_1 + b_2\log a_2+...+b_n\log a_n$, where $|b_i|\leq B_i$ and $1\leq a_i\leq A_i$ are integers, as $A_i,B_i\to \infty$.

Let $A$ be an $a$-letter alphabet. We consider fractional powers of $A$-strings: if $x$ is a $n$-letter string, $x^r$ is a prefix of $xxxx...$ having length $nr$. Let $l$ be a positive integer. Ilie, Ochem and Shallit defined $R(a,l)$ as…

Let $p>3$ be a prime, and let $a$ be a rational p-adic integer with $a\not\equiv 0\pmod p$. In this paper we establish congruences for $$\sum_{k=1}^{(p-1)/2}\frac{\binom ak\binom{-1-a}k}k, \quad\sum_{k=0}^{(p-1)/2}k\binom ak\binom{-1-a}k…

We define the ring of elliptic periods modulo an integer $n$ and give an elliptic version of the AKS primality criterion.